Capacitor, electrical model component

The problem when trying to make an electrical model of the physical or chemical processes in tissue is often that it is not possible to mimic the electrical behavior with ordinary lumped, physically realisable components such as resistors (R), capacitors (C), inductors, semiconductor components, and batteries. Let us mention three examples 1) The constant phase element (CPE), not realizable with a finite number of ideal resistors and capacitors. 2) The double layer in the electrolyte in contact with a metal surface. Such a layer has capacitive properties, but perhaps with a capacitance that is voltage or frequency dependent. 3) Diffusion-controlled processes (see Section 2.4). Distributed components such as a CPE can be considered composed of an infinite number of lumped components, even if the mathematical expression for a CPE is simple. 

To consider the application of microwave irradiation for organic synthesis, the first step is to analyze the reaction components together with their dielectric properties among which of the greatest importance is dielectric constant (cr) sometimes called electric permeability. Dielectric constant (er) is defined as the ratio of the electric permeability of the material to the electric permeability of free space (i.e., vacuum) and its value can derived from a simplified capacitor model (Fig. 1.4). 

The second meaning of the word circuit is related to electrochemical impedance spectroscopy. A key point in this spectroscopy is the fact that any -> electrochemical cell can be represented by an equivalent electrical circuit that consists of electronic (resistances, capacitances, and inductances) and mathematical components. The equivalent circuit is a model that more or less correctly reflects the reality of the cell examined. At minimum, the equivalent circuit should contain a capacitor of - capacity Ca representing the -> double layer, the - impedance of the faradaic process Zf, and the uncompensated - resistance Ru (see -> IRU potential drop). The electronic components in the equivalent circuit can be arranged in series (series circuit) and parallel (parallel circuit). An equivalent circuit representing an electrochemical - half-cell or an -> electrode and an uncomplicated electrode process (-> Randles circuit) is shown below. Ic and If in the figure are the -> capacitive current and the -+ faradaic current, respectively. 

The empirical current-duration relationship is in somewhat better accordance with the hyperbolic model than the exponential, but the exponential model is directly derived from the electric circuit model with a current source supplying an ideal resistor and capacitor in parallel. The empirical current-duration relationship is different for myelinated and naked axons. Also, it must be remembered that the excitation process is nonlinear and not easily modeled with ideal electronic components. 

Usually the two-component circuit is too simple to mimic with sufficient precision the frequency dependence of the variables measured. The three-component model combines features of both the series and parallel models. It may consist of two capacitors and one resistor (dielectrics with bound electric charges) or two resistors and one capacitor (conductors with free charge carriers). Detailed equations can be found in Section 12.2. 

He discussed the three-component electric equivalent circuit with two resistors (one ideal, lumped, physically realizable electronic component one frequency-dependent not realizable) and a capacitor (frequency-dependent) in two different configurations. He discussed his model first as a descriptive model, but later discussed Philippson s explanatory interpretation (extra-/intracellular liquids and cell membranes). 

The electric circnit made np with a capacitor and a resistor in parallel is of great importance thronghout physics becanse it models many relaxation phenomena and imperfections of energy storage. Leaking capacitors, viscoelastic behaviors, permeable barriers, or membranes, in fact all bad (nonideal) energy containers, are modeled by the association of these two components when nsing an eqnivalent electrical circnit. 

It is fonnd from Figure 16.3 that, the dielectric constants of the composites are non-linearly dependent on volume % of BNN. This shows that the constituent capacitors formed by dielectrics fillers and polymer in the composites are not in parallel combination. From Figure 16.3, it is clear that the inverse of dielectric constant cnrve is not in a harmonic pattern, constituent capacitors formed by dielectrics fillers and polymer in the composites is not in series combinatiom One can choose to model composites as having capacitance in parallel (upper bound) or in series (lower bound). In practice, the answer will lie somewhere between the two. Physically, in composites with (0-3) structures which generally conform to special logarithmic equation, the relation assumes the form of Lichteneker and Rother s (Lich-teneker, 1956) more appropriate to composite stractures where the two-component dielectrics are neither parallel nor perpendicular to the electric field that is, the vahd averages are neither arithmetic nor harmonic. 

The waveguide discontinuities shown in Fig. 4.23(a) to Fig. 4.23(f) illustrate most clearly the use of E and H field disturbances to realize capacitive and inductive components. An E-plane discontinuity (Fig. 4.23(a)) can be modeled approximately by a frequency-dependent capacitor. H-plane discontinuities (Fig. 4.23(b) and Fig. 4.23(c)) resemble inductors as does the circular his of Fig. 4.23(d). The resonant waveguide iris of Fig. 4.23(e) disturbs both the E and H fields and can be modeled by a parallel LC resonant circuit near the frequency of resonance. Posts in waveguide are used both as reactive elements (Fig. 4.23(f)) and to mount active devices (Fig. 4.23(g)). The equivalent chcuits of microstrip discontinuities (Fig. 4.23(k) to Fig. 4.23(o)) are again modeled by capacitive elements if the E field is interrupted and by inductive elements if the H field (or current) is interrupted. The stub shown in Fig. 4.23(j) presents a short chcuit to the through transmission line when the length of the stub is A. /4. When the stubs are electrically short (shunt capacitances in the through transmission Hne. 

From the expression in Eq. (19) most forms of equivalent circuit models of piezoelectric elements may be found. The Van Dyke circuit [14] is the simplest, using discrete electrical components combined to approximate the piezoelectric element s behavior. I tis used to represent the electrical irrqtedance about one resonance of a freely suspended piezoelectric element, using a shunt capacitor in parallel with an inductor, resistor, and capacitor placed in series to represent the motional or resonance behavior of the element. A multivibrator Van Dyke model may be formed by adding additional motional legs, each representing another resonance. Since it lacks any explicit treatment of the output force and velocity, it is not especially useful beyond electrical characterization (see... 

The circuit model includes, among others, a resistor, capacitor and inductor. Such component models are available from standard libraries. The model equations are v = R i, i = C der(v) and V =L-der(i), respectively, where der() denotes the time derivative. Each component has two electric pin interfaces (filled and non-filled blue squares in Figure la) that include the voltage v as a potential and current i as a flow variable. Model equations are introduced for connected component (object) interfaces as follows The potential is the same, whereas the sum of flow equals zero according to Kirchhoff s node rule. For example, in case of the circuit model shown by Figure la) v =v = V7 and... 

EIS data are commonly analyzed by fitting them to an equivalent electrical circuit model corresponding to a fuel cell component or components. Most of the circuit elements in the model are common electrical elements such as resistors, capacitors, and inductors. As an example, the electrolyte ohmic resistance can be represented with a resistor. Very few electrochemical cells... 

The same setup is used in impedance spectroscopy. Here, electrode impedance composed of amplitude and phase is measured by putting a sine voltage with varying frequency (V/jy in Fig. 1.6) on the electrode (a DC offset of Vr will exist at Vw) and measuring the resulting current amplitude and phase shift over the frequency. The impedance spectrum of the electrode under test, i.e. Z-w rk versus frequency of Vi, is determined. An electrode model can be extracted from the resulting impedance data. It is usually composed of the prevalent linear components in electrical engineering like resistor and capacitor and components which are defined for the especial use in electrochemistry. One of these elements is introduced in Chap. 8, namely the constant phase element. 

Equivalent circuit (EC) analysis is relatively simple for a circuit containing ideal elements R, C, and L. It may also be carried out for circuits containing distributed elements that can be described by a closed-form equation, such as CPE, semi-infinite, finite length, or spherical diffusion. Many "ideal" resistances and capacitances chosen to represent a real physicochemical system are really nonideal as any resistor has a capacitive component and vice versa. However, for the broad frequency range utilized by UBEIS it is usually adequate to incorporate "ideal" resistors, capacitors, and inductances [29, p. 87]. The type of electrical components in the model and their interconnections... 

Investigations of conductivity of insulating polymers are very difficult due to very low values of the measured currents. Additionally the apparent values of resistivity p or conductivity a are time dependent because the measured current decreases with increasing time after application of the electric field. This effect is due to complex polarization phenomena which occur when a step-voltage is applied to the sample. The resistance and capacitance of the insulating polymer can be represented by equivalent circuits with different serial and parallel combinations of resistors and capacitors in order to model the time (or frequency) dependence of the current. Such simulations can he used for elucidation of the resistance component of the sample. 

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